3.11: Exercises

1. Monochromatic light with a wavelength of \(6000 \unicode{x212b}\) passes through a fast shutter that opens for \(10^{-9}\) sec. What is the subsequent spread in wavelengths of the no longer monochromatic light?
2. Calculate \(\langle x\rangle\), \(\langle x^{\,2}\rangle\), and \(\sigma_x\), as well as \(\langle p\rangle\), \(\langle p^{\,2}\rangle\), and \(\sigma_p\), for the normalized wavefunction \[\psi(x) = \sqrt{\frac{2\,a^{\,3}}{\pi}}\,\frac{1}{x^{\,2}+a^{\,2}}.\] Use these to find \(\sigma_x\,\sigma_p\). Note that \(\int_{-\infty}^{\infty} dx/(x^{\,2}+a^{\,2}) = \pi/a\).
3. Classically, if a particle is not observed then the probability of finding it in a one-dimensional box of length \(L\), which extends from \(x=0\) to \(x=L\), is a constant \(1/L\) per unit length. Show that the classical expectation value of \(x\) is \(L/2\), the expectation value of \(x^{\,2}\) is \(L^2/3\), and the standard deviation of \(x\) is \(L/\sqrt{12}\).
4. Demonstrate that if a particle in a one-dimensional stationary state is bound then the expectation value of its momentum must be zero.
5. Suppose that \(V(x)\) is complex. Obtain an expression for \(\partial P(x,t)/\partial t\) and \(d/dt \int P(x,t)\,dx\) from Schrödinger’s equation. What does this tell us about a complex \(V(x)\)?
6. \(\psi_1(x)\) and \(\psi_2(x)\) are normalized eigenfunctions corresponding to the same eigenvalue. If \[\int_{-\infty}^\infty \psi_1^\ast\,\psi_2\,dx = c,\] where \(c\) is real, find normalized linear combinations of \(\psi_1\) and \(\psi_2\) that are orthogonal to (a) \(\psi_1\), (b) \(\psi_1+\psi_2\).
7. Demonstrate that \(p=-{\rm i}\,\hbar\,\partial/\partial x\) is an Hermitian operator. Find the Hermitian conjugate of \(a = x + {\rm i}\,p\).
8. An operator \(A\), corresponding to a physical quantity \(\alpha\), has two normalized eigenfunctions \(\psi_1(x)\) and \(\psi_2(x)\), with eigenvalues \(a_1\) and \(a_2\). An operator \(B\), corresponding to another physical quantity \(\beta\), has normalized eigenfunctions \(\phi_1(x)\) and \(\phi_2(x)\), with eigenvalues \(b_1\) and \(b_2\). The eigenfunctions are related via \[\begin{aligned} \psi_1 &= (2\,\phi_1+3\,\phi_2) \left/ \sqrt{13},\right.\nonumber\\[0.5ex] \psi_2 &= (3\,\phi_1-2\,\phi_2) \left/ \sqrt{13}.\right.\nonumber\end{aligned}\] \(\alpha\) is measured and the value \(a_1\) is obtained. If \(\beta\) is then measured and then \(\alpha\) again, show that the probability of obtaining \(a_1\) a second time is \(97/169\).
9. Demonstrate that an operator that commutes with the Hamiltonian, and contains no explicit time dependence, has an expectation value that is constant in time.
10. For a certain system, the operator corresponding to the physical quantity \(A\) does not commute with the Hamiltonian. It has eigenvalues \(a_1\) and \(a_2\), corresponding to properly normalized eigenfunctions \[\begin{aligned} \phi_1 &= (u_1+u_2)\left/\sqrt{2},\right.\nonumber\\[0.5ex] \phi_2 &= (u_1-u_2)\left/\sqrt{2},\right.\nonumber\end{aligned}\] where \(u_1\) and \(u_2\) are properly normalized eigenfunctions of the Hamiltonian with eigenvalues \(E_1\) and \(E_2\). If the system is in the state \(\psi=\phi_1\) at time \(t=0\), show that the expectation value of \(A\) at time \(t\) is \[\langle A\rangle = \left(\frac{a_1+a_2}{2}\right) + \left(\frac{a_1-a_2}{2}\right)\cos\left(\frac{[E_1-E_2]\,t}{\hbar}\right).\]

Contributors and Attributions

• Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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